Summary Introduction to Probability for Probability Theory for EOR course
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Course
Probability Theory (EBP014B05)
Institution
Rijksuniversiteit Groningen (RuG)
Book
Introduction to Probability, Second Edition
Summary of the book Introduction to Probability for the course Probability Theory for EOR. The first six chapters are covered in detail in the summary.
Samenvatting Introduction to Probability, Blitzstein & Hwang - Probability Theory
PROBABILITY THEORY FOR EOR | Summary (RUG)
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Rijksuniversiteit Groningen (RuG)
Econometrics And Operations Research
Probability Theory (EBP014B05)
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Probability Theory
Freeke Boerrigter
Chapter 1 & 2 – Probability and Counting
¿ ¿
Naïve definition of probability: Pnaïve = ¿ A∨
¿ S∨¿ ¿
A – the event that A occurs
S – sample space, all possible outcomes
Ac – the event that A does not occur
Pnaïve(Ac) = 1 – Pnaïve(A)
The binomial coefficient formula: ()
n = n!
k ( n−k ) ! k !
where k is picked out of a
total set n.
Choosing the complement: for any nonnegative integers n and k with k ≤ n we
have (nk)=(n−k
n
)
Non-naïve definition of probability – a probability space consists of a sample
space S and a probability function P which takes an event A ⊆ S as input and
returns P(A), a real number between 0 and 1, as output. The function P must
satisfy the following axioms:
1. P(∅) = 0 and P(S) = 1
∞
2. P( ¿ j=1 ¿ ∞ A j ) =∑ P ( A j ) -> this simply means the union of all probabilities
j=1
of A is the sum of all the probabilities of A
Multiplication Rule -> if A does not affect B (independent), then P(A and B) =
P(A)P(B)
Some Events:
P(Ac) = 1 – P(A)
If A⊆ B, then P(A) ≤ P(B)
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
P(S) = P(A) + P(Ac) = 1
Chapter 2 – Conditional Probability
Two events are dependent if knowing one of the outcomes occurred, affects the
other outcome.
The Conditional Probability of A given B is P(A|B)
P ( A ∩ B)
P ( A|B )=
P ( B)
We call P(A) the prior probability of A and P(A|B) the posterior probability
of A
P ( B∨A ) P ( A )
Bayes’ Rule: P ( A|B )=
P( B)
n
The Law of Total Probability (LOTP): P ( B )=∑ P ( B∨ Ai ) P( Ai )
i=1
1
, Simpson’s paradox occurs when groups of data show a particular trend, but
when the data is combined, the trend reverses.
Example: on Saturday you get 7/8 points (87.5%), and your friend gets 2/2 points
(100%). On Sunday, you get 1/2 points (50%), and your friend gets 5/8 points
(62.5%). Both days, your friend has a higher proportion of points however when
you combine the days you have 8/10 points, and your friend has 7/10 points.
This is the paradox, combining groups of data reverses the trend.
Chapter 3 – Random Variables and their Distributions
Probability Mass Function (PMF) is a function that gives the probability that
a discrete random variable is exactly equal to some value.
Bernoulli Distribution only has two outcomes, success or failure. Example is
tossing a coin, getting a head is chance 0.5 and getting a tail is change 1 – p = 1
– 0.5 = 0.5.
Random variable X with parameter p if P(X = 1) = p and P(X = 0) = 1 – p.
Where 0 < p < 1.
X Bern(p)
Binomial Distribution is the outcome of a Bernoulli distribution repeated
multiple times. It has two possible outcomes.
Let X be the number of successes, and n and p be the parameters where n
is a positive integer and 0 < p < 1.
X Bin(n, p)
The PMF of X if X Bin(n, p) is P(X = k) = (nk ) p ( 1− p)
k n−k
Hypergeometric Distribution is used when you want to determine the
probability of obtaining a certain number of successes without replacement from
a specific sample size.
X HGeom(w, b, n) -> w and b come from white and black balls
from the urn
The PMF of X if X HGeom(w, b, n) is P(X = k) =
( k )( n−k )
w b
for
(n)
w+ b
integers k satisfying 0≤ k ≤ w and 0 ≤ n – k ≤ b and P(X = k) = 0.
Discrete Uniform Distribution says that all outcomes are equally likely out of
a finite, nonempty set of numbers.
X DUnif(C) if the chosen number is X
1
The PMF of X if X DUnif(C) is P(X = x) =
¿ C∨¿ ¿
Cumulative Distribution Function (CDF) is a function that gives the
probability that any random variable is exactly equal to some value.
Any function of a random variable is also a random variable itself.
2
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